# How to Find the Equation of a Parallel or Perpendicular Line

In linear algebra, if you are given the equation of a line and the coordinates of a point outside the line, you can find the equation of a parallel line that goes through the given point. You can also determine the equation of a perpendicular line that goes through the given point.

The standard equation form for a line in the xy-coordinate plane is *y = mx + b*, where *m* is the slope of the line (rise over run) and *b* is the y-intercept. A line that is horizontal has a slope of zero, and its equation is simply *y = b*. A line that is vertical has infinite slope, and its equation is *x = a*, where *a* is the x-intercept.

Follow the instructions below to compute parallel and perpendicular lines by hand, or use the calculator on the left to find the equations of the lines.

## Parallel Lines

When two lines are parallel, their slopes are equal. That is, if the equation of the first line is*y = mx + b*, then the equation of a parallel line is

*y = mx + f*, where

*f*is a different y-intercept. When two lines are parallel, the only thing that is different between them is where they cross the y-axis. Here is an example of how to find the equation:

Suppose the equation of a line is

*y*= 3

*x*- 4 and a point has coordinates (10, 13). The equation of the parallel line will be

*y*= 3

*x*+

*f*, where

*f*depends on the coordinates of the point. It is easy to solve for

*f*, just plug in 10 for

*x*and 13 in for

*y*. Then

13 = 3(10) +

*f*

13 = 30 +

*f*

-17 =

*f*

So the equation of the parallel line is

*y*= 3

*x*- 17.

## Perpendicular Lines

When two lines are perpendicular, their slopes are opposite reciprocals. That is, if the equation of the first line is*y = mx + b*, then the equation of a perpendicular line is

*y*= -(1/

*m*)

*x + g*, where

*g*is a different y-intercept. You can compute the equation of a perpendicular line using the same method above. Example:

Suppose the equation of a line is

*y*= 3

*x*- 4 and a point has coordinates (10, 13). The equation of the perpendicular line will be

*y*= -(1/3)

*x*+

*g*, where

*g*depends on the coordinates of the point. If we plug in 10 for

*x*and 13 for

*y*, then

13 = -(1/3)(10) +

*g*

13 = -(10/3) +

*g*

(49/3) =

*g*, or

16.333 =

*g*

So the equation of the perpendicular line is

*y*= -(1/3)

*x*+ (49/3).

© *Had2Know 2010
*